Weak and strong connectivity regimes for a general time elapsed neuron network model

TL;DR

This paper analyzes the long-term behavior of large, fully connected neuron networks modeled by time elapsed dynamics, providing improved estimates in both weak and strong connectivity regimes with and without delay.

Contribution

It offers more accurate estimates of network dynamics compared to previous studies, extending results to non-smooth firing rates and refining spectral analysis methods.

Findings

Enhanced estimates for neuron network behavior in different connectivity regimes

Extension of results to non-smooth firing rate functions

Refined spectral analysis techniques for long-term dynamics

Abstract

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works [7, 8, 6], we establish accurate estimate on the long time behavior of the solutions in the weak and the strong connectivity regime both in the case with and without delay. Our results improve [7, 8] where a less accurate estimate was established and [6] where only smooth firing rates were considered. Our approach combines several arguments introduced in the above previous works as well as a slightly refined version of the Weyl's and spectral mapping theorems presented in [13, 4].